Generic singularities for 2D pressureless flows

In this paper, we consider the Cauchy problem for pressureless gases in two space dimensions with the generic smooth initial data (density and velocity). These equations give rise to singular curves, where the mass has a positive density with respect to the 1-dimensional Hausdorff measure. We observe that the system of equations describing these singular curves is not hyperbolic. For analytic data, local solutions are constructed by using a version of the Cauchy-Kovalevskaya theorem. We then study the interaction of two singular curves in the generic position. Finally, for a generic initial velocity field, we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed.